[Very often, an index number used in an economic model has been constructed in two or more stages. If the two stage procedure gives the same answer as a single stage procedure, then Vartia calls the index number formula "consistent in aggregation." Paasche and Laspeyres indexes have this consistency in aggregation property, but these index number formulae are consistent only with very restrictive functional forms for the underlying aggregator (i.e., utility or production) function. The present paper shows that the class of superlative index number formulae has an approximate consistency in aggregation property, where a superlative index number formula is one which is consistent with a flexible functional form for the underlying aggregator function. The paper also contains some empirical examples which both illustrate the main theorem and also indicate that the chain principle for constructing index numbers is preferable to the fixed base method. Finally, the paper proves some theorems about the class of pseudo-superlative index numbers.]

This paper provides necessary and sufficient conditions for it to be optimal to base decisions on estimates of the parameters that characterize a decision problem (e.g., profit maximization with an estimated price elasticity of demand). We show that the separation of parameter estimation from decision making generally yields lower utility than an integrated approach which takes account of estimation uncertainty. We evaluate the decision in the parameter estimation method and show that the resulting utility loss can be substantial. MANY ACTUAL DECISIONS are based on statistical estimates of parameters that help to characterize the decision environment. For example, a firm maximizing the expected utility of profit might find that its input and output decisions depend on unknown parameters of its demand function. Econometric estimates of such parameters might then be derived and utilized in making these decisions. The first purpose of this paper is to rigorously investigate whether it is correct to make decisions in this manner; in general, it is not. The second purpose is to investigate the decis'ion bias in decisions based on commonly employed parameter estimates. We will determine, for example, whether a price setting monopolist is mistakenly setting prices too high or too low when he bases his pricing decision on the maximum likelihood estimate of his demand equation. Finally, we provide a detailed numerical example to show that basing decisions on conventional parameter estimates can lead to large losses of utility. In Section 2, we introduce all notation, explain the procedure commonly used when basing decisions on values of unknown underlying parameters, and exhibit the decision-theoretic correct alternative procedure. When the optimal decisions under these two procedures are identical, we call the proper. We use the term summary value to refer not only to standard parameter estimates but to any single substituted for an unknown parameter in order to make decisions. This generalized concept is necessary because a that is appropriate for making decisions, in a sense defined below, need not have any of the properties of conventional parameter estimators. In Section 3, we derive under general assumptions necessary and sufficient conditions for the existence of proper values that are independent of the decision maker's utility function, U( ). This independence restriction is